Fat Tails, Fourth Moments, and GARCH Volatility Estimates
Summary
The document discusses whether GARCH variance estimates remain useful when returns have heavy tails. It relays the concern that if a return distribution has a tail exponent near three, its fourth moment may be infinite, making the variance of squared returns—and thus the precision of variance estimates—problematic. It contrasts variance-based estimates with mean absolute deviation, which may be more stable but less responsive to sharp volatility shocks.
The author describes a simulation using Student-t samples and observes substantial uncertainty in estimated standard deviation relative to mean absolute deviation. They then qualify the concern: financial returns exhibit clustered, conditional volatility rather than being independent and identically distributed, which may make variance estimates more informative in practice. The discussion raises useful questions but supplies no detailed simulation results, formal derivation, or tested trading method. Its claims should be read as exploratory; heavy tails do not by themselves establish that conditional variance models are unusable.
Key ideas
- Heavy tails can make fourth-moment-based uncertainty measures unreliable.
- Variance estimates may react strongly to extreme observations while remaining statistically imprecise.
- Mean absolute deviation can offer greater stability but may respond less to volatility shocks.
- Volatility clustering means financial returns are not well described by an independent, identically distributed sample.
- The document presents an exploratory argument rather than a validated comparison of forecasting methods.
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Full text
# GARCH, Infinite Variance of Variance # GARCH, Infinite Variance of Variance Taleb criticised GARCH, saying that for daily stock returns with tail exponent ~3 the 4th moment doesn't exist and so Var of Var doesn't exist. The whole concept of conditional variance breaks apart. Can you please clarify the practical consequences of that statement? As far as I understand it means - it's not possible to measure current variance with good precision (GARCH, EMA), etc). What would be correct approach? I see two: - Make estimation stable by removing extreme events. Use GARCH, but always buy far out of money put/call options (they usually have around zero ITM probability and cost pennies, so it's a viable strategy). - Use GARCH, EMA with MeanAbsDev instead of Variance. Problem - I found that GARCH with MeanAbsDev has lower LLH than GARCH with Variance, I think it because Variance detects "shock" better. N. Taleb, "Statistical Consequences of Fat Tails", page 51: > For silver, in 46 years 94 percent of the kurtosis came from one single observation. We cannot use standard statistical methods with financial data. GARCH does not work because we are dealing with squares. The variance of the squares is analogous to the fourth moment. We do not know the variance. UPDATE 1: I think the problem - not possible to measure variance with high precision. Experiment - distribution of estimated std and mean abs dev in 10k trials of `sample(StudentT(0,1,2.7), 1000)`, charts with var convergence. STD has huge uncertainty in estimation. Seems like: - STD sensitive to shock, but low precision. - MeanAbsDev high precision but low shock sensitivity. UPDATE 2: I missed important detail. Stock prices have conditional variance (clusters of volatility), not i.i.d. And so, the variance measurements are more reliable. It may be ok to use Variance, GARCH, etc.
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